Stable Harmonic Analysis and Stable Transfer

TL;DR

This paper proves the existence of stable transfer operators in the context of the Langlands program, using stable Paley--Wiener theorems, advancing the understanding of functoriality for p-adic groups.

Contribution

It establishes the existence of continuous stable transfer operators for various function spaces assuming a hypothesis on the local Langlands correspondence, introducing new stable Paley--Wiener theorems.

Findings

Stable Paley--Wiener theorem for Harish-Chandra Schwartz functions

Stable tempered characters span a weak-* dense subspace of stable tempered distributions

Existence of stable transfer operators assuming local Langlands hypothesis

Abstract

Langlands posed the question of whether a local functorial transfer map of stable tempered characters can be interpolated by the transpose of a linear operator between spaces of stable orbital integrals of test functions. These so-called stable transfer operators are intended to serve as the main local ingredient in Beyond Endoscopy, his proposed strategy for proving the Principle of Functoriality. Working over a local field of characteristic zero and assuming a hypothesis on the local Langlands correspondence for p-adic groups, we prove the existence of continuous stable transfer operators between spaces of stable orbital integrals of Harish-Chandra Schwartz functions, test functions, and K-finite test functions. This is achieved via stable Paley--Wiener theorems for each of the three types of function spaces. The stable Paley--Wiener theorem for Harish-Chandra Schwartz functions is new and includes the result that stable tempered characters span a weak-* dense subspace of the space of stable tempered distributions, a result previously unknown for p-adic groups.